The complex numbers definition can be given as any number which can be represented in the form: x + iy, where ‘x’ is known as the real part of the complex number and iy is known as the imaginary part of the complex number. But you must know in complex number definition that x and y are both Real Numbers and i is the imaginary unit which satisfies the property i2 = -1
Complex PlaneBack to Top
We use the concept of the complex plane which allows a geometric interpretation of complex numbers. Now let us look at the complex numbers. The numbers which are written in the form of ( a + ib), where we have a and b as the Real Numbers and I = root (-1) is the representation of the complex number. The Set of all complex numbers are represented by C. For any complex number, z = a + ib, we have a as the real part of z, which is written as Re( z) and b is the imaginary part of z, written as Im( z).
(4 + 3i), here 4 is real part of the complex number and 3 is the imaginary part of the complex number.
We say that the number is a pure real number, when we have the Im (z) = 0, such as 4 , 7, root(3) are all the pure real numbers. On the other hand, we have the pure Imaginary Number, if the real part of the complex number i.e. a = 0. Thus, we write Re (z) = 0 2i, root (7) I, -3i/2 are all pure imaginary numbers.
Modulus of the complex number z is represented as |z| = root ( a^2 + b^2).
Suppose we have a complex number z = 3 + 4 I, then the modulus of z = root ( 9 + 16) = root (25) = 5.
Complex ConjugateBack to Top
Now we will learn about complex conjugate, where we say that if the complex number z is represented as a + bi then its mathematical complex conjugate is represented as Ż = a – ib. To find the complex conjugates of the complex number z= 2 + 3i, we will simply write it as conj (Z) = 2-3i
Another important thing to be remembered is that if the complex number z and its conjugate are added, then the imaginary part of the two numbers cancels out, as one is a positive number and the other is a negative. Let us try to add the above given z and its conjugate, then we have:
Z + conjugate ( Z) = 2 + 3i + 2 – 3i
= 4 + 3i – 3i
= 4 Ans
Let us look at a special situation of finding the conjugate:
If we have z = i^3, the to find its conjugate we precede as follows:
We know that z = i^3 = -I [ as we know that i^2 = 1]
So z = 0 – i
Thus we write conjugate ( z) = 0 + i = i Ans